On a fixed point theorem of Greguš

Abstract

We consider two selfmaps T and I of a closed convex subset C of a Banach space X which are weakly commuting in X, i.e.‖TIx−ITx‖≤‖Ix−Tx‖   for   any   x   in   X,and satisfy the inequality‖Tx−Ty‖≤a‖Ix−Iy‖+(1−a)max{‖Tx−Ix‖,‖Ty−Iy‖}for all x, y in C, where 0<a<1. It is proved that if I is linear and non-expansive in C and such that IC contains TC, then T and I have a unique common fixed point in C.